Derivative Definition and 143 Discussions

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.
Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.
The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.

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  1. AwesomeTrains

    Question about the derivation of the energy momentum tensor

    Hey I'm trying to follow the derivation given here: Homework Statement As it says in the pdf: "Based on Noether's theorem construct the energy-momentum tensor for classical electromagnetism from the above Lagrangian. L=-1/4...
  2. Nader AbdlGhani

    What is the Higher Derivative of a function?

    I'm relatively new to calculus and I have a new chapter in my study which is on the Implicit Function, Implicit Differentiation and Higher Derivatives of a function, the problem is I don't understand the meaning of a 2nd or 3rd or whatever the higher derivative of a function is, what I know is...
  3. T

    Splitting a Derivative

    After doing a couple courses in physics as well as calculus and differential equations, I was starting to wonder about splitting a derivate, such as ## \frac{dy}{dx} ##, into separate pieces ##dy## and ##dx##. I know we've never done it in calculus or differential equations because it isn't...
  4. N

    Distance formula maximization problem

    Homework Statement At 9 P.M. an oil tanker traveling west in the ocean at 18 kilometers per hour passes the same spot as a luxury liner that arrived at the same spot at 8 P.M. while traveling north at 23 kilometers per hour. If the "spot" is represented by the origin, find the location of the...
  5. M

    I'm curious what anyone gets for part d

    Homework Statement After a wild ride on a windsurfer, you head to a lifeguard tower to fly a kite. You stand on the tower, which is 3 meters above the ground and release your kite. You let out 10 meters of string and the kite begins moving according to the following equation: y(t) = (0.9...
  6. N

    Why is the first derivative velocity?

    Just something that has been bugging me. Can someone bestow why the first derivative is velocity and the second derivative is acceleration. I want to conceptually understand this. Thank you
  7. J

    Calc BC derivative problem with trig and double angle -- Help please

    Homework Statement Find f'(x) if f(x) = 8^(sin^2(3x)) Hint: you will need to use the double angle formula for trig functions and your answer should only have one trig function in it. Homework Equations if y=a^u then y' = ln a * a^u * du sin(2x) = 2sinxcosx The Attempt at a Solution We're...
  8. J

    How to find the equation of this tangent?

    Mod note: Thread moved from Precalc section Homework Statement F(x)=sqrt(-2x^2 +2x+4) 1.discuss variation of f and draw (c) 2.find the equation of tangent line to (c) that passes through point A(-2,0) The Attempt at a Solution I solved first part I found the domain of definition and f'(x)...
  9. TyroneTheDino

    Finding the distance from origin to a tangent line

    Homework Statement Find the distance between the origin and the line tangent to ##x^\frac{2}{3}+y^{\frac{2}{3}}=a^{\frac{2}{3}}## at the point P(x,y) Homework Equations [/B] Distance= ##\frac{\left |a_{0}+b_{0}+c \right |}{\sqrt{a^{2}+b^{2}}}## The Attempt at a Solution To begin I find...
  10. N

    What version of the definition of derivative

    If my understanding is correct the definition of a derivative is lim h->0 (f(x+h)-f(x))/h However, I've also seen this used: lim x->c (f(x)-f(x))/(x-c) are these both considered valid definition for the derivative or does the derivative have to tend towards zero? I am a bit confused because I...
  11. Titan97

    Question on Mean Value Theorem & Intermediate Value Theorem

    Homework Statement for ##0<\alpha,\beta<2##, prove that ##\int_0^4f(t)dt=2[\alpha f(\alpha)+\beta f(\beta)]## Homework Equations Mean value theorem: ##f'(c)=\frac{f(b)-f(a)}{b-a}## The Attempt at a Solution I got the answer for the question but I have made an assumption but I don't know if...
  12. N

    Confused on these two derivative problems

    Homework Statement a. k(t) = (sqrt(t+1))/(2t+1) b. y = (3^(x^2+1))(ln(2)) The Attempt at a Solution For the first problem, I know I use the quotient rule for derivatives (L)(DH)-(H)(DL)/((L)^2) which would go to: ((2t+1)(1/(2sqrt(t+1)) - (sqrt(t+1))(2))/((2t+1)^2) I get stuck here, maybe...
  13. haael

    How does the Kalman filter calculate derivatives?

    Suppose we have a Kalman filter. We have a position sensor, for example GPS. We use the filter to estimate position. However in all examples I see higher derivatives in the state vector: speed, acceleration and sometimes jerk. There is no sensor that calculates these values directly, so they...
  14. A

    Velocity Derivative of a Sinusoidal Wave (Counter-Intuitive)

    What's the matter: So, I think I have some skills when it comes to differentiation after taking calculus 2 last semester, but when it starts to intertwine with physics, and interpreting physical phenomenon through equations, It appears I could use some help. Anyway, the problem that I got hung...
  15. R

    Tension on a Rope Deflected by a Pulley: Differentials

    Hi all, first post here. I'm a junior Physics/Math double major at UMass Amherst, playing with some problems over the summer. I'll get right into it. A rope with constant tension T is deflected through the angle 2\theta_{0} by a smooth, fixed pulley. What is the force on the pulley? It is...
  16. Chrono G. Xay

    'Wheel-like' Mathematics (Modulating Trig Functions?)

    As part of a personal musicology project I found myself with the mathematical model of a geometry which utilizes the equation a*(a/b)sin(pi*x) The only problem with this is that I need to take the integral from -1/2 <= x <= 1/2, and according to Wolfram Alpha no such integral exists. I can...
  17. P

    Foundations What's a good book for last year of A levels Maths?

    I'm looking for a book to self-study this summer before my last course of Bachillerato (A Levels or High School in other countries). My performance in Mathematics has only improved and I've just been given the maximum mark this last term (10/10 or A+). This has motivated me a lot to keep...
  18. R

    Derivatives and Linear transformations

    Let G be a non-empty open connected set in Rn, f be a differentiable function from G into R, and A be a linear transformation from Rn to R. If f '(a)=A for all a in G, find f and prove your answer. I thought of f as being the same as the linear transformation, i.e. f(x)=A(x). Is this true?
  19. leafjerky

    Deriving formulas used for vectors in physics.

    Hey guys and gals, I'm taking an online physics course just to kind of learn the basics before I take the real thing this summer. The course is from OpenYale for those interested ( Anyways, the professor was talking about some formulas for finding ##\vec{r}(t) = r(i(cos\omega t) +...
  20. M

    Calculus Derivative Terminology

    I have attached an image of a function that I fit to a scatter plot, and I would like to know if there is a term for the point on the function at which the slope transitions from being less than -1 to greater than -1. I have highlighted this point approximately in yellow...
  21. Prof. 27

    Linear Ordinary Differential Equation: Definition

    Homework Statement The website says this: "It is Linear when the variable (and its derivatives) has no exponent or other function put on it. So no y2, y3, √y, sin(y), ln(y) etc, just plain y (or whatever the variable is). More formally a Linear Differential Equation is in the form: dy/dx +...
  22. AdityaDev

    Finding y=f(x)

    Homework Statement Given two curves y=f(x) passing through (0,1) and ##g(x)=\int\limits_{-\infty}^xf(t)dt## passing through (0,1/n). The tangents drawn to both curves at the points with equal abscissae intersect on the x-axis. Find y=f(x). Homework Equations None The Attempt at a Solution...
  23. B

    Derivative of metric with respect to metric

    I'm hoping someone can clarify for me, I have seen the following used: \frac{\partial}{\partial g^{ab}}\left( g^{cd} \right) = \frac{1}{2} \left( \delta_a^c \delta_b^d + \delta_b^c \delta_a^d\right) I understand the two half terms are used to account for the symmetry of the metric tensor...
  24. D

    Richardson Extrapolation with 3 steps?

    Homework Statement [/B] use richardson extrapolation to estimate the first derivative y=ln(x), x=5 using steps of 2, 1, 0.5. Four decimal points. obtain true relative error for the last estimate and comment on its value. Homework Equations [/B] deriv ln(x)=1/x The Attempt at a Solution I...
  25. Albo1125

    Using the derivative of the formula of the number of nuclei

    Hi all, I have a question concerning the derivative of the formula of the number of nuclei. I hope I've posted this in the right section, I'm new here :P. Anyway, in the question, the given values are: At a certain time t, there is an amount of radioactive Br-82. The activity A is 7.4*1014 Bq...
  26. T

    Maximize volume of a box

    Suppose someone gives you a rectangular sheet of length a and width b (so b ≤ a). You make a topless box by cutting out a square with length x out of each corner and folding up the sides. How should you cut the sheet so as to maximize the volume of your box?
  27. S

    Natural log differentiation question

    Homework Statement f(x) = 1/ln (10-x) -- I would assume it to be a fairly simple equation, but I am screwing it up Homework Equations What is f'(x)? The Attempt at a Solution f'(x) = (ln (10-x))^-1 = -(ln (10-x))^-2 * -1 * 1/(10-x) -- 2 negatives cancel out = 1/(10-x)...
  28. Ondina

    Complex derivative of x: ((x)^(1/x))'

    <<Moderator note: Remember that filling in the complete homework template is mandatory in the homework forums. This thread has not been deleted due to containing relevant replies.>> 1. Homework Statement ((x)^(1/x))' Homework Equations This probably isn't overly dificult, but it has got me...
  29. S

    New Calc student w/ a derivative question

    Homework Statement Hello all, thank you for the help in advance. It's a two-sided derivative problem, for lack of a better term, and I appreciate all hints or help. If we have a function y so that y=bx for all x<0, and y= x^2-13x for all x> or = 0, for what value of b is y differentiable at...
  30. N

    Taking the time derivative of a curl

    Is the time derivative of a curl commutative? I think I may have answered this question... Only the partial time derivative of a curl is commutative? The total time derivative is not, since for example in cartesian coordinates, x,y,and z can themselves be functions of time. In spherical and...
  31. C

    Kinetic Energy Time Derivative

    Homework Statement So the first part asks to prove the time derivative of kinetic energy is dT/dt=F dot product v which I did not problem. but then the second part of the problem asks to prove that if the mass is changing with time then the time derivative of d(mT)/dt=F dot product m and I'm...
  32. powerof

    Symmetry in second order partial derivatives and chain rule

    When can I do the following where ##h_{i}## is a function of ##(x_{1},...,x_{n})##? \frac{\partial}{\partial x_{k}}\frac{\partial f(h_{1},...,h_{n})}{\partial h_{m}}\overset{?}{=}\frac{\partial}{\partial h_{m}}\frac{\partial f(h_{1},...,h_{n})}{\partial x_{m}}\overset{\underbrace{chain\...
  33. D

    Proving the fundamental theorem of calculus using limits

    Would it be a legitimate (valid) proof to use an \epsilon-\delta limit approach to prove the fundamental theorem of calculus? i.e. as the FTC states that if f is a continuous function on [a,b], then we can define a function F: [a,b]\rightarrow\mathbb{R} such that F(x)=\int_{a}^{x}f(t)dt Then F...
  34. N

    For f(x) = abs(x^3 - 9x), does f'(0) exist

    Homework Statement For f(x) = abs(x^3 - 9x), does f'(0) exist? The Attempt at a Solution [/B] The way I tried to solve this question was to find the right hand and left hand derivative at x = 0. Right hand derivative = (lim h--> 0+) f(h) - f(0) / h = (lim h--> 0+) abs(h^3 - 9h) / h...
  35. T

    A little confused about integrals

    I learned that integrals are finding the area under a curve. But I seem to be a little confused. Area under the curve of the derivative of the function? Or area under the curve of the original function? If an integral is the area under a curve, why do we even have to find the anti derivative...
  36. S

    Queries regarding Inflection Points in Curve Sketching

    Homework Statement Let A be a set of critical points of the function f(x). Let B be a set of roots of the equation f''(x)=0. Let C be a set of points where f''(x) does not exist. It follows that B∪C=D is a set of potential inflection points of f(x). Q 1: Can there exist any inflection points...
  37. Diana Dobleve

    Logarithmic derivative question

    Homework Statement 1) I am having trouble with the questions, "Use the logarithmic derivative to find y' when y=((e^-x)cos^2x)/((x^2)+x+1) Homework Equations (dy/dx)(e^x) = e^x (dy/dx)ln(e^-x) = -x ? The Attempt at a Solution First I believe I put ln on each set of terms (Though I don't know...
  38. T

    Finding the derivative of a revenue function

    Homework Statement The revenue function for a product is r = 8x where r is in dollars and x is the number of units sold. the demand function is q = -1/4p + 10000 where q units can be sold when selling price is p. what is dr/dp? Homework Equations r=pq The Attempt at a Solution I substituted...
  39. W

    Proof showing that if F is an antiderivative of f, then f must be continuous.

    Homework Statement Show that if F is an antiderivative of f on [a,b] and c is in (a,b), then f cannot have a jump or removable discontinuity at c. Hint: assume that it does and show that either F'(c) does not exist or F'(c) does not equal f(c). 2. The attempt at a solution I attempted a proof...
  40. admbmb

    Conceptual trouble with derivatives with respect to Arc Length

    Hi, So I'm working through a bunch of problems involving gradient vectors and derivatives to try to better understand it all, and one specific thing is giving me trouble. I have a general function that defines a change in Temperature with respect to position (x,y). So for example, dT/dt would...
  41. thegreengineer

    Derivative as a rate of change exercise

    Homework Statement A police car is parked 50 feet away from a wall. The police car siren spins at 30 revolutions per minute. What is the velocity the light moves through the wall when the beam forms angles of: a) α= 30°, b) α=60°, and c) α=70°? This is the diagram...
  42. T

    Help with Wronskian Equation

    Homework Statement W(t) = W(y1, y2) find the Wronskian. Equation for both y1 and y2: 81y'' + 90y' - 11y = 0 y1(0) = 1 y1'(0) = 0 Calculated y1: (1/12)e^(-11/9 t) + (11/12)e^(1/9 t) y2(0) = 0 y2'(0) = 1 Calculated y2: (-3/4)e^(-11/9 t) + (3/4)e^(1/9 t) Homework Equations W(y1, y2) = |y1...
  43. A

    Kinematics Acceleration question

    Homework Statement Suppose a can, after an initial kick, moves up along a smooth hill of ice. Make a statement concerning its acceleration. A) It will travel at constant velocity with zero acceleration. B) It will have a constant acceleration up the hill, but a different constant acceleration...